Multi-Dimensional, Compressible Viscous Flow on a Moving Voronoi Mesh

Numerous formulations of finite volume schemes for the Euler and Navier-Stokes equations exist, but in the majority of cases they have been developed for structured and stationary meshes. In many applications, more flexible mesh geometries that can dynamically adjust to the problem at hand and move with the flow in a (quasi) Lagrangian fashion would, however, be highly desirable, as this can allow a significant reduction of advection errors and an accurate realization of curved and moving boundary conditions. Here we describe a novel formulation of viscous continuum hydrodynamics that solves the equations of motion on a Voronoi mesh created by a set of mesh-generating points. The points can move in an arbitrary manner, but the most natural motion is that given by the fluid velocity itself, such that the mesh dynamically adjusts to the flow. Owing to the mathematical properties of the Voronoi tessellation, pathological mesh-twisting effects are avoided. Our implementation considers the full Navier-Stokes equations and has been realized in the AREPO code both in 2D and 3D. We propose a new approach to compute accurate viscous fluxes for a dynamic Voronoi mesh, and use this to formulate a finite volume solver of the Navier-Stokes equations. Through a number of test problems, including circular Couette flow and flow past a cylindrical obstacle, we show that our new scheme combines good accuracy with geometric flexibility, and hence promises to be competitive with other highly refined Eulerian methods. This will in particular allow astrophysical applications of the AREPO code where physical viscosity is important, such as in the hot plasma in galaxy clusters, or for viscous accretion disk models.Comment: 26 pages, 21 figures. Submitted to MNRA

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders. Part I: The one-dimensional case

International audienceOne of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in the one-dimensional Lagrangian gas dynamics framework. After briefly recalling how to derive Lagrangian forms of the 1D gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS, and relies on two different techniques: either a particular definition of the local approximation of the acoustic impedances arising from the approximate Riemann solver, or an additional time step constraint relative to the cell volume variation. Then, making use of the work presented in [89, 90, 22], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. This paper is the first part of a series of two. The whole analysis presented here is extended to the two-dimensional case in [85], and proves to fit a wide range of numerical schemes in the literature, such as those presented in [19, 64, 15, 82, 84]

Transition in a numerical model of contact line dynamics and forced dewetting

We investigate the transition to a Landau-Levich-Derjaguin film in forced dewetting using a quadtree adaptive solution to the Navier-Stokes equations with surface tension. We use a discretization of the capillary forces near the receding contact line that yields an equilibrium for a specified contact angle $\theta_\Delta$ called the numerical contact angle. Despite the well-known contact line singularity, dynamic simulations can proceed without any explicit additional numerical procedure. We investigate angles from $15^\circ$ to $110^\circ$ and capillary numbers from $0.00085$ to $0.2$ where the mesh size $\Delta$ is varied in the range of $0.0035$ to $0.06$ of the capillary length $l_c$. To interpret the results, we use Cox's theory which involves a microscopic distance $r_m$ and a microscopic angle $\theta_e$. In the numerical case, the equivalent of $\theta_e$ is the angle $\theta_\Delta$ and we find that Cox's theory also applies. We introduce the scaling factor or gauge function $\phi$ so that $r_m = \Delta/\phi$ and estimate this gauge function by comparing our numerics to Cox's theory. The comparison provides a direct assessment of the agreement of the numerics with Cox's theory and reveals a critical feature of the numerical treatment of contact line dynamics: agreement is poor at small angles while it is better at large angles. This scaling factor is shown to depend only on $\theta_\Delta$ and the viscosity ratio $q$. In the case of small $\theta_e$, we use the prediction by Eggers [Phys. Rev. Lett., vol. 93, pp 094502, 2004] of the critical capillary number for the Landau-Levich-Derjaguin forced dewetting transition. We generalize this prediction to large $\theta_e$ and arbitrary $q$ and express the critical capillary number as a function of $\theta_e$ and $r_m$. An analogy can be drawn between $r_m$ and the numerical slip length.Comment: This version of the paper includes the corrections indicated in Ref. [1

Numerical simulations of neutron stars in general relativistic hydrodynamics

In dieser Arbeit wurden ein Computerprogramm zur Simulation der Vakuum Einsteingleichungen erweitert um die allgemeinen relativistischen Hydrodynamik Gleichungen zu l¨osen. Diese wurden benutzt um numerische Neutronensternen zu simulieren, insbesondere um den Kollapse eines Neutronensternes zu untersuchen und um Gravitationswellen von Bin¨arsystemen zu extrahieren. Die verwendeten numerischen Methoden werden beschrieben und an Testf¨allen validiert. Die Implementation der HRSC Methode wurde am Shocktube validiert. Bei der Simulation eines stabilen Sterns konnte Konvergenz gezeigt werde. Die erwartete Oszillationsfrequenz des Sterns in radialer Richtung stimmt mit der Literatur ¨uberein. Weiterhin wurde ein bewegter simuliert und Konvergenz gezeigt. Der Kollapse eines instabilen Neutronensternes und das daraus entstehende schwarze Loch wurde mit der Raumzeit einem einzelnen schwarzen Loches bei unter gleicher Eichung verglichen. Es wurde gezeigt, dass beide zur gleichen L¨osung tendieren die mit der analytischen L¨osung in guter N¨aherung bereinstimmt. Das Verschwinden der Materie konnte durch die benutzte shift-Bedingung erkl¨art werden. Weiterhin wurden bin¨are Neutronenstern Systeme betrachtet die sich anf¨anglich in einem quasi-Equilibrium befindet und sich n¨aherungsweise auf Kreisbahnen bewegt. F¨ur diese Simulationen konnten wir w¨aerend des Einspiralens Konvergenz und sehr gute Massenerhaltung zeigen. Der finale Stern kollabiert zu einem schwarzen Loch mit einer Akkretionsscheibe. Der Effekt von unterschiedlichen Eichbedingungen und verschiedenen Zustandsgleichungen auf die Simulation und auf die extrahierten Gravitationswellen wurden betrachte

Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: From first-order to high-orders

One of the main issues in the field of numerical schemes is to ally robustness with accuracy. Considering gas dynamics, numerical approximations may generate negative density or pressure, which may lead to nonlinear instability and crash of the code. This phenomenon is even more critical using a Lagrangian formalism, the grid moving and being deformed during the calculation. Furthermore, most of the problems studied in this framework contain very intense rarefaction and shock waves. In this paper, the admissibility of numerical solutions obtained by high-order finite-volume-scheme-based methods, such as the discontinuous Galerkin (DG) method, the essentially non-oscillatory (ENO) and the weighted ENO (WENO) finite volume schemes, is addressed in this Lagrangian gas dynamics framework. To this end, we first focus on the one-dimensional case. After briefly recalling how to derive Lagrangian forms of the gas dynamics system of equations, a discussion on positivity-preserving approximate Riemann solvers, ensuring first-order finite volume schemes to be positive, is then given. This study is conducted for both ideal gas and non ideal gas equations of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Mie-Grüneisen (MG) EOS. It enables us to derive time step conditions ensuring the desired positivity property, as well as L 1 stability of the specific volume and total energy over the domain. Then, making use of the work presented in [74, 75, 15], this positivity study is extended to high-orders of accuracy, where new time step constraints are obtained, and proper limitation is required. This whole analysis is finally applied to the two-dimensional case, and shown to fit a wide range of numerical schemes in the literature, such as the GLACE scheme [12], the EUCCLHYD scheme [55], the GLACE scheme on conical meshes [8], and the LCCDG method [72]. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. Finally, let us emphasize that even if this paper is concerned with purely Lagrangian schemes, the theory developed is of fundamental importance for any methods relying on a purely Lagrangian step, as ALE methods or non-direct Euler schemes